Week 2a Lecture_Utility

# Week 2a Lecture_Utility - Week 2 – Part I Utility...

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Unformatted text preview: Week 2 – Part I Utility Preferences - A Reminder • x y: x is weakly preferred to y. • x y: x is strictly preferred to y. • x ∼ y: x and y are equally preferred. • We assume that a preference is complete (reflexive) and transitive. ~ Utility Functions • If a preference relation that is additionally continuous can be represented by a continuous utility function . • Continuity means that small changes to a consumption bundle cause only small changes to the preference level. Utility Functions • A utility function U(x) represents a preference relation if and only if: x’ x” U(x’) >= U(x”) We get the following for free: x’ x” U(x’) > U(x”) x’ x” U(x’) = U(x”). ~ ~ & ~ Utility Functions • Utility is an ordinal (i.e. ordering) concept. • E.g . if U(x) = 6 and U(y) = 2 then bundle x is strictly preferred to bundle y. But x is not preferred three times as much as is y. • If we can find one utility representation, then we have infinitely many. Utility Functions • U(x 1 ,x 2 ) = x 1 x 2 (2,3) (4,1) (2,2). • Define V = U 2 . • Then V(x 1 ,x 2 ) = x 1 2 x 2 2 and V(2,3) = 36 > V(4,1) = V(2,2) = 16 so again (2,3) (4,1) (2,2). • V preserves the same order as U and so represents the same preferences. ~ ~ Utility Functions • If – U is a utility function that represents a preference relation and – f is a strictly increasing function, • then V = f(U) is also a utility function representing . ~ ~ Utility Functions & Indiff. Curves • An indifference curve contains equally preferred bundles. • Equal preference ⇒ same utility level....
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Week 2a Lecture_Utility - Week 2 – Part I Utility...

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